on the state space {0, 1, 2, 3, ...}. The model is a type of birth–death process. We write ρ = λ/(c μ) for the server utilization and require ρ < 1 for the queue to be stable. ρ represents the average proportion of time which each of the servers is occupied (assuming jobs finding more than one vacant server choose their servers randomly).

If the traffic intensity is greater than one then the queue will grow without bound but if server utilization ρ=λcμ<1{\displaystyle \rho ={\frac {\lambda }{c\mu }}<1} then the system has a stationary distribution with probability mass function[4][5]

The customer either experiences an immediate exponential service, or must wait for k customers to be served before their own service, thus experiencing an Erlang distribution with shape parameter k + 1.[10]

In a processor sharing queue the service capacity of the queue is split equally between the jobs in the queue. In the M/M/c queue this means that when there are c or fewer jobs in the system, each job is serviced at rate μ. However, when there are more than c jobs in the system the service rate of each job decreases and is cμn{\displaystyle {\frac {c\mu }{n}}} where n is the number of jobs in the system. This means that arrivals after a job of interest can impact the service time of the job of interest. The Laplace–Stieltjes transform of the response time distribution has been shown to be a solution to a Volterra integral equation from which moments can be computed.[11] An approximation has been offered for the response time distribution.[12][13]

In an M/M/c/K queue (sometimes known as the Erlang–A model[1]:495) only K customers can queue at any one time (including those in service[4]). Any further arrivals to the queue are considered "lost". We assume that K ≥ c. The model has transition rate matrix

The average number of customers in the system is E[N]=λμ+π0ρ(cρ)c(1−ρ)2c!{\displaystyle \mathbb {E} [N]={\frac {\lambda }{\mu }}+\pi _{0}{\frac {\rho (c\rho )^{c}}{(1-\rho )^{2}c!}}}[16] and number of average response time for a customer is E[T]=1μ+π0ρ(cρ)cλ(1−ρ)2c!{\displaystyle \mathbb {E} [T]={\frac {1}{\mu }}+\pi _{0}{\frac {\rho (c\rho )^{c}}{\lambda (1-\rho )^{2}c!}}}[16] .